Causal Inference in Metric Spaces via Distance Profiles

Modern interventions increasingly produce object-valued outcomes, such as distributions, covariance matrices, networks, and trees. Existing causal frameworks often exploit special geometry such as one-dimensional Wasserstein quantiles or geodesic Fréchet means, whereas metric statistics studies random objects without confounding adjustment. We propose the profile causal effect $Δ(ω,t)$, which describes where and at what scale a treatment shifts probability mass, including changes in dispersion, tails, or structure that leave mean-level summaries unchanged. Formally, $Δ(ω,t)=\Pr\{d(ω,Y(1))\le t\}-\Pr\{d(ω,Y(0))\le t\}$ contrasts the potential-outcome probabilities of the same anchor-centered metric ball and requires no structure beyond the metric. On grids fixed by a deterministic or auxiliary design, we combine cross-fitted doubly robust scores with high-dimensional Gaussian approximation and the multiplier bootstrap to obtain simultaneous confidence bands and a global test under regularity conditions. The bands bound treatment-induced changes in ball and shell probabilities simultaneously and, at fixed anchors, extend to all radii. A protocol guides the choice of anchors and radii. Across the experiments, the global test detects concentration effects that mean-quantile and Fréchet-mean summaries miss. Its empirical size is near or below the nominal level, even under confounding, where naive metric tests are substantially oversized.